Hybrid Physical-Data Modeling Approach for Surface Scattering Characteristics of Low-Gloss Black Paint

Abstract

This study presents a hybrid BRDF modeling framework combining a five-parameter physical model with a (20,20,20) Multilayer Perceptron (MLP) model network to address the critical challenge of accurate grazing-angle prediction for low-gloss black coatings (SB-3A, Z306, PNC). While the baseline parametric model achieves <5% RMSE at θ_i ≤ 60°, its inability to capture shadowing effects leads to >1.2 RMSE at 80° incidence. The proposed MLP model-enhanced solution reduces these high-angle errors to <0.012 RMSE while maintaining <5-min computational efficiency. Comprehensive validation shows the framework’s universality across materials with apparent anisotropy indices (ASI) of 0.465–30.26. The work demonstrates that neural networks can optimally compensate for missing physics in traditional models without sacrificing interpretability, offering immediate industrial value for aerospace coating analysis.

1. Introduction

Low-gloss black paints, known for their extremely low reflectance (typically under 2%), are essential functional materials for suppressing stray light in advanced optical systems [1,2,3]. These paints are widely used in key components like baffles and light traps within precision optical instruments, such as space telescopes and LiDAR systems. By absorbing stray radiation outside the intended optical path, they significantly improve background noise suppression in high-sensitivity detection systems. This enhancement boosts the system’s signal-to-noise ratio and detection reliability [4,5].

The bidirectional reflectance distribution function (BRDF) is a fundamental parameter for characterizing the reflective properties of material surfaces. Over the past half-century, BRDF modeling methodologies have evolved into four major paradigms. Physics-based models grounded in microfacet theory, such as the Cook-Torrance model, derive macroscopic scattering behavior by statistically describing the distribution of surface normal, offering theoretical rigor but requiring a priori assumptions about surface statistics [6,7,8]. Numerical models based on Maxwell’s equations, such as the finite-difference time-domain (FDTD) method, provide precise solutions for electromagnetic field distributions at the nanoscale, but are limited by high computational complexity, making them impractical for engineering applications [9,10,11,12,13,14]. Empirical models, including Lambertian, Phong, and the Five-Parameter Model, strike a balance between physical fidelity and data fitting accuracy through parameter coupling, and have thus become the mainstream choice in optical engineering [15,16,17]. Among them, the Five-Parameter Model stands out for its moderate complexity and broad applicability, effectively capturing both diffuse and specular reflection characteristics with five adjustable parameters. In recent years, researchers have sought to enhance the Five-Parameter Model by refining surface distribution or masking functions to extend its application in engineering fields [18,19,20,21,22,23,24,25,26]. With the advent of artificial intelligence, data-driven BRDF modeling approaches, such as neural networks and machine learning algorithms, have gained attention in remote sensing. However, these methods heavily rely on large volumes of high-quality experimental data for training, and the acquisition cost for low-reflectance materials remains prohibitively high, limiting their scalability [27,28,29]. It is noteworthy that most existing BRDF models have been developed based on high-reflectance materials such as metals, glass, and vegetation, making their underlying assumptions and optimization strategies less suited to the unique light–matter interaction mechanisms of low-gloss paints. Specifically, scholars from the Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, proposed a modified microfacet model for low-gloss black paints, incorporating high-order polynomial correction factors to achieve high-accuracy BRDF modeling under large incident angles [30,31]. While this approach improves adaptability at extreme angles, it still faces challenges such as limited material generalization and reduced physical interpretability of the high-order terms.

Modeling the BRDF of low-gloss black paints presents two fundamental challenges: the material’s strong light absorption invalidates classical specular scattering assumptions, while coupled shadowing and multiple scattering effects create complex azimuthal dependencies. Traditional optimization methods struggle with the nonlinear parameter space, often converging to local optima, while data-driven approaches face overfitting with limited samples. To address these issues, we propose a hierarchical framework combining a physics-based five-parameter baseline model (optimized via gradient descent) with a Multilayer Perceptron (MLP) model that learns residual scattering patterns. This hybrid approach explicitly decouples the BRDF into interpretable empirical components and neural network corrections, achieving superior accuracy in modeling azimuthal apparent anisotropy and high-incidence-angle behavior (θ_i > 60°).

2. Materials and Methods

2.1. BRDF Modeling for Optical Black Paints

2.1.1. Gradient Descent-Optimized Parameter Fitting for a Five-Parameter Model

Wang et al. modified the key terms of the Fresnel reflection function and shadowing function based on the Torrance-Sparrow model, proposing a five-parameter semi-empirical statistical model [24]. This model employs five fitting parameters $({\mathrm{k}}_b{,\mathrm{k}}_{\mathrm{r}}{,\mathrm{k}}_d,\mathrm{a},\mathrm{b})$ and four empirical parameters $({\sigma }_p,{\sigma }_{\mathrm{r}},u_p,{\upsilon }_p)$ to characterize the scattering properties of material surfaces, capable of simultaneously describing both diffuse and specular reflection characteristics. Multiple researchers have further refined its key components—such as the surface distribution function, diffuse reflection term, and shadowing function—to adapt it for modeling material scattering properties in the field of spectral remote sensing [32]. The Five-Parameter Model is widely used in remote sensing for high-reflectance materials such as vegetation and man-made surfaces. However, for low-reflectance coatings, strong absorption suppresses specular peaks, and microstructural shadowing induces azimuthal apparent anisotropy. These effects degrade model accuracy, especially at high incident angles (θ_i > 60°), revealing its limited ability to handle angle-dependent scattering nonlinearities (e.g., shadowing-induced local maxima/minima in BRDF profiles) in low-gloss surfaces.

BRDF mathematical description is expressed as $f({\theta }_i,{\varphi }_i,{\theta }_r,{\varphi }_r)$, as shown in Equation (1), and Angle definitions in the Five-Parameter Model are shown in Figure 1.

$${\displaystyle \frac{f({\theta }_i,{\varphi }_i,{\theta }_r,{\varphi }_r)}{\cos {\theta }_{\mathrm{i}}}}={\displaystyle \frac{k_b\cdot k_r^2\cos \alpha }{1+(k_r^2-1)\cos \alpha }}\exp [b\cdot {(1-\cos \gamma )}^a]{\displaystyle \frac{G({\theta }_i,{\theta }_r,{\varphi }_r)}{\cos {\theta }_i\cos {\theta }_r}}+{\displaystyle \frac{k_d}{\cos {\theta }_i}}$$

(1)

In the Equation (1), ${\theta }_i$ and ${\theta }_r$ denote the incident and reflected zenith angles, respectively; ${\varphi }_i$ and ${\varphi }_r$ represent the incident and reflected azimuth angles; ($k_b$, $k_d$, $k_r$, a, b) are fitting parameters, k_b represent a scaling coefficient that controls the overall strength of the specular reflection component. It represents the normalized magnitude of the specular lobe, k_r is a shape parameter associated with the specular reflection. It determines the angular spread (width) of the specular lobe, k_d represent the diffuse reflection coefficient, a is an exponential parameter of the distribution correction, b is a scaling factor of the distribution correction that controls the amplitude of the angular modulation; $\alpha$ is the angle between the facet’s normal direction (n) and the normal direction of the macroscopic object surface (z), and can be expressed by the following Equation (2); $\gamma$ is the angle between the incident light and the facet’s normal direction, can be expressed by the following Equation (3); G represents the shadowing function, which accounts for the masking and shadowing effects of microfacets, and can be expressed by the following Equations (4) and (5).

$$\cos \alpha ={\displaystyle \frac{\cos {\theta }_i+\cos {\theta }_r}{2\cos \gamma }}$$

(2)

$${\cos }^2\gamma ={\displaystyle \frac{1}{2}}(\cos {\theta }_i\cos {\theta }_r+\sin {\theta }_i\sin {\theta }_r\cos {\varphi }_r+1)$$

(3)

$$G({\theta }_i,{\theta }_r,{\varphi }_r)={\displaystyle \frac{1+\frac{{\omega }_p\left|\tan {\theta }_p^i\tan {\theta }_p^r\right|}{1+{\sigma }_r\tan {\gamma }_p}}{(1+{\omega }_p{\tan }^2{\theta }_p^i)(1+{\omega }_p{\tan }^2{\theta }_p^r)}}$$

(4)

$${\omega }_{\mathrm{p}}(\alpha )={\sigma }_p(1+{\displaystyle \frac{u_p\sin \alpha }{\sin \alpha +{\upsilon }_p\cos \alpha }})$$

(5)

In the Equation (4), ${\theta }_p^i$, ${\theta }_p^r$ and ${\gamma }_p$ are the spherical projections of ${\theta }_i$, ${\theta }_r$ and $\gamma$, respectively. Several empirical parameters (${\sigma }_p$, ${\sigma }_r$, $u_p$, ${\upsilon }_p$) in the Five-Parameter Model are assigned fixed values recommended in previous literature (Reference [24]). These values have been validated across various remote sensing applications and are sufficient to characterize typical surface roughness effects. Fixing them reduces the optimization dimensionality and improves model robustness without significantly affecting overall fitting accuracy. This study adopts the following empirical values: (${\sigma }_p=0.0136$, ${\sigma }_{\mathrm{r}}=0.0136$, $u_p=9.0$, ${\upsilon }_p=1.0$). Based on spherical trigonometry formulas, the expressions for parameters ${\theta }_p^i$, ${\theta }_p^r$ and ${\gamma }_p$ in the shadowing function are given by the following Equations (6)–(8):

$$\tan {\theta }_p^i=\tan {\theta }_i{\displaystyle \frac{\sin {\theta }_i+\sin {\theta }_r\cos {\varphi }_r}{2\sin \alpha \cos \gamma }}$$

(6)

$$\tan {\theta }_p^r=\tan {\theta }_i{\displaystyle \frac{\sin {\theta }_r+\sin {\theta }_i\cos {\varphi }_r}{2\sin \alpha \cos \gamma }}$$

(7)

$$\tan {\gamma }_p={\displaystyle \frac{\left|\cos {\theta }_i-\cos \gamma \right|}{2\sin \alpha \cos \gamma }}$$

(8)

In practice, conventional algorithms such as ordinary least squares (OLS), simulated annealing algorithm (SAA), and particle swarm optimization (PSO) are commonly employed for parameter identification based on measured datasets [33,34]. In recent years, with breakthroughs in deep neural network training techniques, gradient descent (GD)-based iterative optimization methods have demonstrated significant advantages in BRDF modeling. Research results indicate that, on the same dataset, the fitting error of gradient descent-based optimization algorithms is consistently lower than that of other optimization approaches [35].

The key parameters for gradient descent optimization of the Five-Parameter Model are shown in

Table 1. Key Parameters of the GD.

Key Parameters	Value
Loss Function	MSE
Initial Parameters	(0.5, 0.5, 0.5, 0.5, 0.5)
Learning Rate	0.01
Max Iterations	500
Tolerance	1 × 10−6

below. This study employs a linear MSE fitting, with the loss function defined as Equation (9):

$$\mathrm{loss}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{(f_i^{true}-f_i^{pred})}^2}$$

(9)

where $f_i^{true}$, $f_i^{pred}$ represent the measured BRDF and model-predicted BRDF value, respectively, for the N is number of data points.

2.1.2. Data-Driven MLP Model

To address the limitations of traditional empirical models in capturing complex, apparent anisotropic scattering behaviors of low-gloss black paints, this study integrates a data-driven component based on the MLP model. The Multilayer Perceptron (MLP) model, a classic feedforward neural network (FNN), consists of an input layer, one or more hidden layers, and an output layer, with information propagation between layers achieved through fully connected weight matrices. Each neuron in the MLP model applies a nonlinear activation function, enabling the network to learn complex data mappings. Through backpropagation algorithms and gradient descent optimization, the MLP model trains its weights in supervised learning tasks to minimize the loss function. Compared to other neural network architectures, the MLP model imposes no specific requirements on data formats and can directly process flattened feature vectors, resulting in lower data preparation costs. Its simple computational graph and short gradient propagation paths enable faster and more stable optimization. Furthermore, through fully connected layers, the MLP model implicitly learns global correlations among features, exhibiting higher tolerance to local noise or partial feature missing [36,37]. These advantages motivate our adoption of the MLP model as the data-driven BRDF prediction model in this study. This section details the model architecture, training strategy, and implementation specifics.

The MLP model adopted in this work is a feedforward neural network designed to approximate the mapping between geometric configuration parameters and residual BRDF values. This includes 4 input layers, 3 hidden layers and 1 linear output layer. The input layer takes four geometrical variables: incident zenith angle ${\theta }_i$, incident azimuth angle ${\varphi }_i({\varphi }_i=0)$, reflected zenith angle ${\theta }_r$, and reflected azimuth angle ${\varphi }_r$, with the output being the predicted BRDF value. The MLP model architecture consists of 3 hidden layers, each containing twenty neurons. The network is trained to output the residual correction term $\Delta f_{MLP}$, which is subsequently added to the prediction of the Five-Parameter Model.

The final BRDF prediction, $f_{\mathrm{h}ybrid}({\theta }_i,{\varphi }_i,{\theta }_r,{\varphi }_r)$, is thus given by the following:

$$f_{\mathrm{h}ybrid}({\theta }_i,{\varphi }_i,{\theta }_r,{\varphi }_r)=f_{FP}({\theta }_i,{\varphi }_i,{\theta }_r,{\varphi }_r)+\Delta f_{MLP}$$

(10)

where $f_{FP}({\theta }_i,{\varphi }_i,{\theta }_r,{\varphi }_r)$, $\Delta f_{MLP}$ represent the Five-Parameter model BRDF, Residual MLP BRDF.

To ensure reproducibility and robustness, the training procedure is configured as follows, with the key parameters detailed in

Table 2. Key Training Parameters of the MLP model.

Key Training Parameters	Value
Optimizer	Trainlm (Levenberg–Marquardt)
Loss Function	logarithmic residuals
Early stopping	max_fail = 20
Epochs	1000
Data Split	70% training, 15% validation, 15% test

. The BRDF dataset includes measurements over four discrete incident angles (0°, 30°, 60°, 80°), each with full scattering azimuth and zenith scans, yielding a total of 12,996 samples. For each incident angle, 70% of the data is randomly assigned to the training set, 15% to the validation set and 15% to the testing set. The MLP model is trained to minimize the log-scale residual between predicted and measured BRDF values. The MLP model optimizer employs Trainlm (Levenberg–Marquardt algorithm) with dynamic learning rate adaptation, configured for 1000 epochs. The loss function utilizes logarithmic residuals, and early stopping is implemented with max_fail = 20 validation checks. The algorithm’s default regularization is implemented through the damping factor μ (mu), which provides adaptive implicit regularization, distinct from traditional L1/L2 weight penalties. The loss function is defined as Equation (9). It is worth noting that all theoretical modeling and plotting presented in this paper were conducted utilizing MATLAB R2023a.

Unlike pure data-driven models that often lack interpretability, the proposed MLP model is trained not to directly predict the BRDF, but to correct the deficiencies of a physically grounded baseline model. This residual learning strategy has twofold benefits: (1) Physical consistency: By anchoring predictions to the Five-Parameter Model, the MLP model outputs remain within physically plausible bounds, avoiding unbounded oscillations in poorly sampled regions. (2) Focused learning: The MLP model concentrates on modeling angular regions where the baseline model underperforms, such as high-angle shadowing zones or off-specular azimuthal peaks. This targeted compensation leads to improved generalization, especially in cases of limited data (e.g., low-reflectance materials where dense sampling is costly).

2.2. BRDF Experimental Testing

Bidirectional Reflectance Distribution Function (BRDF) is a fundamental parameter characterizing surface scattering properties (Figure 2), mathematically defined as the ratio of differential reflected radiance (dL_r) to incident irradiance (dE_i) per unit solid angle, unit: sr⁻¹, expressed as Equation (11):

$$f({\theta }_i,{\varphi }_i,{\theta }_r,{\varphi }_r,\lambda )={\displaystyle \frac{d{L}_r({\theta }_i,{\varphi }_i,{\theta }_r,{\varphi }_r,\lambda )}{d{E}_i({\theta }_i,{\varphi }_i,\lambda )}}$$

(11)

${\theta }_i$ and ${\theta }_r$ denote the incident and reflected zenith angles, respectively; ${\varphi }_i$ and ${\varphi }_r$ represent the incident and reflected azimuth angles; $\lambda$ denote the measurement wavelength (unit: nm).

The working principle of the scatterometer is as follows: prior to testing the sample, a standard white diffuse reflector is selected as the reference panel, whose BRDF is $\rho /\pi$, where $\rho$ is the hemispherical reflectance of the reference panel. Assuming an incident angle of 30°, the response voltage of the reference panel is V_R, and the response voltage of the test sample is V_S. The BRDF $f_{rs}$ of the sample surface can then be expressed by the following Equation (12):

$$f_{rs}={\displaystyle \frac{V_S({\theta }_i,{\varphi }_i,{\theta }_r,{\varphi }_r)\cos {30}^{\circ }}{V_R({30}^{\circ },{\varphi }_i,{\theta }_r,{\varphi }_r)\cos {\theta }_r}}\cdot {\displaystyle \frac{\rho }{\pi }}$$

(12)

To investigate the scattering characteristics of Low-gloss black paint surfaces, this study employed a high-precision scatterometer developed by IOF to measure the BRDF of the surface. The instrument exhibits a BRDF measurement sensitivity of <10⁻⁷ sr⁻¹, a measurement repeatability (RMS) better than 2%, and a measurement uncertainty better than 3%. In our experimental setup, we employed the hemispherical coordinate system (θ_r ∈ [−90°, 90°], Φ_r ∈ [−90°, 90°]) for data acquisition. This approach is commonly adopted in engineering applications due to its practical advantages for mechanical design and measurement efficiency. The sample testing parameters were as follows: the incident light wavelength was 640 nm, the incident angles tested were 0°, 30°, 60°, and 80°, the scattering angle range was −85° to 85° with a 1° interval, and the azimuth angle range was −90° to 90° with a 10° interval. Figure 3 presents the experimental setup, including (a) a photograph of the three black paints (SB-3A, Z306, PNC) and (b) the actual BRDF scatterometer system used for measurements.

A comprehensive symbol table (

Table 3. Comprehensive symbol table.

Geometric Parameters			Physical Quantities
Symbol	Definition	Unit	Symbol	Definition	Unit
$\alpha$	the angle between z direction and the surface normal n	-	ASI	Azimuthal Sensitivity Index (Equation (13))	-
G	Shadowing function	-	dEi	Differential incident irradiance	W/m2
$\gamma$	the angle between the incident light ray and the surface normal	°	dLr	Differential reflected radiance	W/m2/Ω
${\gamma }_p$	the spherical projections of $\gamma$	°	$f({\theta }_i,{\varphi }_i,{\theta }_r,{\varphi }_r)$	Mathematical description of BRDF	sr−1
${\theta }_i$	Incident zenith angle	°	$f_{FP}$	Five-Parameter model BRDF	sr−1
${\theta }_r$	Reflected zenith angle	°	$f_{\mathrm{h}ybrid}$	Final model BRDF	sr−1
${\theta }_p^i$	the spherical projections of ${\theta }_i$	°	$f_i^{\mathrm{pred}}$	Model-predicted BRDF value	sr−1
${\theta }_p^r$	the spherical projections of ${\theta }_r$	°	$\Delta f_{MLP}$	MLP model BRDF residual term	sr−1
${\varphi }_i$	Incident azimuth angle	°	$f_{rs}$	BRDF of the sample surface	sr−1
${\varphi }_r$	Reflected azimuth angle	°	VS	Response voltage of the sample	V
${\omega }_{\mathrm{p}}(\alpha )$	Surface Distribution Function	-	VR	Response voltage of the reference panel	V
Model Parameters			$\rho$	the hemispherical reflectance of the reference panel	-
a	Exponential parameter of the distribution correction	-	$\lambda$	Measurement wavelength	nm
b	Scaling factor of the distribution correction	-
${\sigma }_p$	Fixed engineering parameter	-
${\sigma }_{\mathrm{r}}$	Fixed engineering parameter	-
$u_p$	Fixed engineering parameter	-
${\upsilon }_p$	Fixed engineering parameter	-
$k_b$	Specular reflection scaling coefficient	-
$k_d$	Diffuse reflection coefficient	-
$k_r$	Specular lobe shape parameter	-

) has been clarified all parameters and variables, with detailed definitions as follows:

3. Results

3.1. Model Fitting Results for SB-3A

3.1.1. Analysis of Scattering Characteristics

Figure 4 presents the three-dimensional BRDF projection distributions of the low-gloss black paint SB-3A at different incident angles (0°, 30°, 60°, 80°). The horizontal and vertical axes represent horizontal component ($\sin {\theta }_r\cos {\varphi }_r$) and vertical component ($\sin {\theta }_r\sin {\varphi }_r$) of the scattering vector, respectively, while the color intensity indicates a logarithmic encoding of the BRDF magnitude. Figure 4a exhibits a relatively uniform distribution, indicating minimal variation in the scattering characteristics of the low-gloss black paint across different azimuthal directions, a behavior typical of surfaces dominated by diffuse reflection. This suggests near-Lambertian scattering properties. Figure 4b reveals an asymmetric distribution with noticeable azimuthal deviation, demonstrating that as the incident angle increases, the low-gloss black surface begins to exhibit apparent anisotropic scattering characteristics. Figure 4c,d further intensify this apparent anisotropy, particularly at high incident angles, where scattered energy distinctly concentrates in specific directions. This phenomenon is attributed to enhanced specular reflection and shadowing effects.

The proposed Azimuthal Sensitivity Index (ASI) is defined as a metric to quantify the apparent anisotropic scattering intensity of a surface under fixed mounting conditions. It characterizes the strength of observed azimuthal variation in BRDF, calculated as the normalized difference between maximum and minimum reflectance values at a fixed incident angle, as expressed in Equation (13). This empirical descriptor captures the composite effect of surface topography and material properties on directional scattering, providing a practical measure for comparing coating performance in engineering applications where components remain stationary relative to the optical system. The ASI value directly indicates material apparent anisotropy strength—higher ASI values correspond to stronger directional dependence, with typically isotropic materials exhibiting ASI ≤ 1.5. The ASI distribution of low-gloss black paint SB-3A is shown in Figure 5. The results indicate that when θ_i ≤ 30°, the ASI generally approaches 1 (ranging from 0.730 to 1.25), suggesting weak apparent anisotropy and approximately isotropic scattering characteristics. As the incident angle increases to 60°, the maximum ASI rises to 12.62, demonstrating gradually enhanced apparent anisotropic behavior. When θ_i ≥ 60°, the ASI exhibits an order-of-magnitude leap, with the maximum difference reaching 65-fold (ASI: 0.674–12.62 at θ_i = 60°; ASI: 0.465–30.26 at θ_i = 80°). This pronounced angular dependence, particularly the threshold behavior beyond 60° incidence, provides a critical validation metric for BRDF model accuracy in capturing high-incident angle scattering phenomena. The ASI metric thus serves as both a quantitative apparent indicator and a valuable tool for evaluating surface optical properties in material characterization applications.

$$ASI={\displaystyle \frac{M\mathrm{ax}[BRDF({\theta }_i,{\theta }_r,{\varphi }_r\left)\right]-M\mathrm{in}[BRDF({\theta }_i,{\theta }_r,{\varphi }_r\left)\right]}{Mean[BRDF({\theta }_i,{\theta }_r,{\varphi }_r\left)\right]}}$$

(13)

3.1.2. Gradient-Descent Optimized Five-Parameter Model Fitting Results

As is illustrated in Figure 6, the gradient descent optimization exhibited rapid convergence, with the loss function curve showing steep reduction within the first 100 iterations. The algorithm achieved convergence at 450 iterations, reaching the preset tolerance of 10⁻⁶ through implemented early stopping, which effectively prevented unnecessary computational overhead. As is illustrated in

Table 4. Fitted Parameters of GD-Optimized Five-Parameter Model.

kb	kd	kr	a	b	RMSE
−0.0239	0.0398	0.2782	0.5327	0.4429	0	0.0173
					30	0.0197
					60	0.0288
					80	1.2066

, The Five-Parameter BRDF model shows good fitting results at lower angles but struggles at steep angles. The RMSE values are 0.017 at 0°, 0.019 at 30°, and 0.0288 at 60°, indicating accurate modeling for most cases. However, at 80°, the error jumps to 1.2, mainly due to missing shadowing effects in the model. The results suggest the model works well for moderate angles but needs improvement for grazing angles.

3.2. Hybrid Model Fitting Results

3.2.1. Residual Results Analysis

Figure 7 demonstrates the 3D residual distribution across the full angular domain. The 3D scatter plot visualization of the Five-Parameter Model residuals, highlighting the top 5% high-magnitude residuals (threshold = 95th percentile of absolute values), reveals distinct angular-dependent performance characteristics.

Table 5. Five-Parameter Model Fitting Residuals.

Incident Angle θi (°)	Residual (sr−1)
0	0.0075
30	0.0079
60	0.25
80	26.42

states the model demonstrates satisfactory fitting accuracy for incidence angles below 60° (RMSE < 0.03), with residuals showing relatively uniform distribution across scattering and azimuth angles. However, significant fitting limitations emerge at grazing incidence conditions—while localized high residuals first appear at edge azimuth angles (e.g., θ_i = 60°, θ_r = 80°, φ_r = ±90°), their spatial coverage and magnitude exhibit progressive expansion with increasing incidence angle. Particularly at 80° incidence, the maximum residual values escalate substantially (from 0.25 at 60° to 26.42 at 80°), with anomalous regions spreading across wider angular ranges. These systematic patterns clearly indicate the model’s deteriorating performance in capturing shadowing/masking effects under grazing illumination, suggesting that the current five-parameter formulation requires targeted improvements for high-incident-angle scenarios. The residual distribution trends strongly advocate for incorporating additional physical terms or angular correction factors specifically optimized for θ_i > 60° cases in future model development.

3.2.2. Hybrid Model Fitting Results

To address the high residuals in the five-parameter BRDF model (particularly at grazing incidence angles and edge azimuthal directions), MLP model is employed to nonlinearly correct the systematic errors. The results, as illustrated in Figure 8, demonstrate the MLP model’s effectiveness in reducing these discrepancies: the absolute residuals decreased dramatically from 26.42 to 0.0461 (

Table 6. Hybrid Model Fitting RMSE and Residuals.

Incident Angle θi (°)	RMSE	Residuals (sr−1)
0°	2.576 × 10−4	0.0461
30°	2.993 × 10−4	0.0049
60°	1.19 × 10−3	0.0020
80°	1.178 × 10−2	0.0019

). Notably, at 60° incidence angle, the RMSE dropped from 0.0288 to 0.00119, and for 80° incidence, the RMSE was reduced from 1.2 to just 0.017. Although the MLP model also achieved sub-1% errors at small incidence angles, it should be emphasized that the purely data-driven MLP model lacks physical interpretability. Therefore, its primary value lies in correcting large incident angle errors where the parametric model fails, particularly in grazing angle scenarios (θ_i >60°) where complex shadowing and scattering effects dominate.

3.2.3. Ablation Experiment Results

To evaluate the performance of different network architectures, a systematic study was conducted on the RMSE of neural networks with varying depths. The results (Figure 9) demonstrate that shallow networks ((10), (20)) exhibit poor performance at incidence angles greater than 60° (RMSE: 0.6357~1.188), while the three-layer (20,20,20) network achieves optimal performance (0.01178 at 80°). The network architecture (30,30,30) achieves superior accuracy with RMSE below 0.001 at θ_i ≤ 60°. However, this configuration proves impractical for real-world applications due to prohibitively long execution times exceeding 20 min, rendering it computationally inefficient despite its precision advantages. Comprehensive analysis reveals that the (20,20,20) architecture strikes the best balance between complexity and accuracy, reducing errors at 80° by 99% compared to the [10] network.

3.3. Validation of Model Predictive Capability

To further validate the generalization capability of the proposed modeling framework, this section presents a comparative analysis of the scattering characteristics of Z306 and PNC, with reference to the previously discussed SB-3A results. Figure 10 and Figure 11 show the tangent plane projections of BRDF distributions for Z306 and PNC across four incident angles (θ_i = 0°, 30°, 60°, 80°). Comparative analysis reveals consistent scattering patterns across all incident conditions: (1) All materials transition from near-isotropic scattering at θ_i = 0° to strongly anisotropic distributions at θ_i = 80°; (2) Similar trends of increasing azimuthal variation with increasing θ_i; BRDF dynamic ranges show similar variation patterns with θ_i. Notably, The BRDF distributions of SB-3A and Z306 exhibit similar patterns but with intensity differences, where SB-3A demonstrates overall higher reflectance values compared to Z306 across all measurement conditions. For PNC coating, distinct scattering characteristics are observed: under normal incidence (θ_i = 0°), a prominent specular reflection peak is present, while at other incident angles (θ_i = 30°, 60°, 80°), the material shows relatively uniform BRDF distributions with reduced directional dependence. At θ_i = 30°, subtle intensity variations are observed between the materials: SB-3A maintains higher overall BRDF values compared to both Z306 and PNC coatings. The comparative analysis reveals subtle but discernible variations in both the distribution patterns and magnitude of BRDF among the three materials. Despite these differences, all three coatings maintain characteristically low reflectance properties typical of optical black surfaces, with BRDF values consistently below 1% across the most measured angular range.

To verify the generalization capability of the hybrid model, we applied it to characterize low-gloss black paint material (Z306 and PNC). The results, as illustrated in

Table 7. Fitting Results for Z306 and PNC Low-gloss Black Paint.

Black Paints	Incident Angle θi (°)	Five-Parameter Model		RMSE (Hybrid Model)	ASI
		Fitting Parameters	RMSE
Z306	0	kb = −0.0349	0.0210	4.152 × 10−3	2.3901
	30	kd = 0.0506	0.0250	4.782 × 10−3	2.4053
	60	kr = 0.2796	0.0382	1.384 × 10−2	8.5256
	80	a = 0.5326, b = 0.4428	1.2831	1.867 × 10−2	24.9666
PNC	0	kb = 0.0114	0.0061	2.459 × 10−2	0.6070
	30	kd = 0.0046	0.0068	5.240 × 10−3	0.8935
	60	kr = 0.2761	0.0159	7.656 × 10−3	5.9640
	80	a = 0.5327, b = 0.4426	0.2484	1.230 × 10−2	32.0912

, demonstrate that the Five-Parameter Model achieves high accuracy (RMSE generally below 5%) for incident angles ≤ 60° across the tested black paints (Z306 and PNC), indicating strong generalization for moderate angles. However, at extreme angles (80°), the model’s performance degrades significantly (e.g., RMSE up to 1.2831 for Z306), highlighting its limitations under such conditions. By an MLP-based hybrid model, the fitting precision improves substantially, with RMSE consistently below 2% for all incident angles. This confirms the effectiveness of machine learning in enhancing optical property predictions, particularly in high-angle scenarios.

4. Discussion

This study establishes a comprehensive framework for modeling the apparent anisotropic scattering properties of low-gloss black coatings (SB-3A, Z306, PNC). The five-parameter baseline model demonstrates reliable performance (<5% RMSE) for incidence angles ≤ 60° across all three materials, confirming its utility for conventional optical applications. However, fundamental limitations emerge at grazing angles (θ_i > 60°), where the model’s inability to account for surface-topography-induced shadowing effects results in significant errors (>1.2 RMSE at 80°). The proposed (20,20,20) MLP model architecture addresses these limitations through data-driven learning while maintaining computational efficiency. Key achievements include: (1) Uniform high-angle error reduction: 1.2066 → 0.01178 (SB-3A), 1.2831 → 0.018 (Z306), and 0.2484 → 0.0123 (PNC) RMSE at 80°. (2) Real-time compliance: <5 min execution time versus >20 min for (30,30,30).

The MLP model’s success stems from its capacity to implicitly learn missing higher-order scattering terms while preserving the parametric model’s interpretable foundation. This hybrid approach proves particularly effective for materials (PNC: ASI = 32.0912). covering most of aerospace coating. Future developments will focus on theory-guided network optimization, specifically: Developing angle-adaptive neuron allocation strategies, Implementing spectral sensitivity modules for multiband applications.

It is pertinent to acknowledge that the adopted modeling framework prioritizes empirical accuracy over strict physical adherence—a deliberate design choice justified by practical engineering requirements. The fixed azimuth measurement configuration, while operationally efficient, inherently limits direct experimental verification of Helmholtz reciprocity. Nevertheless, the model demonstrates exceptional empirical fidelity within the operational domain of interest, providing reliable predictions for aerospace coating applications where practical accuracy supersedes theoretical completeness.

5. Conclusions

This study demonstrates that the hybrid (20,20,20) MLP model-enhanced framework successfully bridges the critical gap in BRDF modeling for low-gloss black coatings, achieving unprecedented accuracy-efficiency balance by reducing 80° RMSE from >1.2 to <0.012 across SB-3A, Z306, and PNC (ASI ≈ 32) while maintaining <5-min runtime compliant. The results fundamentally advance BRDF modeling by proving that neural networks optimally compensate for missing shadowing physics at grazing angles (θ_i > 60°) without replacing interpretable parametric foundations. Future developments will prioritize microfacet-theory-guided pruning for speed gains, spectral adaptation modules, and building on the established framework of aerospace coating scenarios.